[rd11235]

I don't have a great answer. Most modern descriptions are shallow and/or unclear. My favorite discussions were actually in Werbos's original papers.

A nice overview was Backpropagation through time: what it does and how to do it, 1990. The rule itself is stated very clearly there, but without proof. The proof can be found in Maximizing long-term gas industry profits in two minutes in lotus using neural network methods, 1989 (which I believe was copied over from his earlier thesis, which I could never find a copy of).